Forbidden Symmetries
Frank A. Farris
T
he crystallographic restriction, as it
applies to patterns in the plane, tells
us that a pattern invariant under two
linearly independent isometries cannot have 5-fold symmetry. And yet the
pattern in Figure 1 seems to have translational
symmetry in two directions as well as rotational
symmetry through 72◦ . To see what I mean, start
with the wheel shape at the point labeled A and
---→
-notice translational symmetries along vectors AB
----→
-and AC; then rotate the image 72◦ about A, so that
B goes to C, and see that the image is unchanged,
apparently in violation of the crystallographic
restriction. How can this be?
A moment’s thought can break the illusion: If
the pattern really enjoyed translational symmetry
---→
along vector AB, then we could rotate 72◦ clockwise about B and move A to C. Alas, angle ∠ABC
is an unfortunate 54◦ , so B is not truly a translate
of A.
The purpose of this paper is to show that
an effort to construct functions known not to
exist may on occasion produce interesting frauds.
Our method produces a family of Harald Bohr’s
quasiperiodic functions, which may well remind
readers of the quasicrystals that have been much
in the news since Daniel Shechtman won the Nobel
Prize in Chemistry in 2011 [1].
The term quasicrystal has an interesting history, as explained by Senechal in the Notices [6].
Diffraction patterns found by Shechtman in 1982
Frank A. Farris is associate professor of mathematics at
Santa Clara University. His email address is ffarris@
scu.edu.
DOI: http://dx.doi.org/10.1090/noti904
1386
Figure 1. Wallpaper with 5 -fold symmetry?
displayed 5-fold symmetry and so fell outside the
mathematical categories commonly accepted as
encompassing all possible crystalline structures.
Our quasiperiodic functions are a different sort
of object altogether. Unlike the structures studied
by crystallographers, which are idealized as sets
of isolated points in space, these are smooth functions that have honest 5-fold symmetry about a
single point and come so close to having translational symmetry that they can easily fool the eye,
provided we select a suitable translation.
The techniques developed to create these functions may offer more interest beyond creating
fraudulent images such as Figure 1. We show how
Notices of the AMS
Volume 59, Number 10
to construct functions with 3-fold symmetry and
how the technique breaks down when we try to
change 3 to 5.
an = anP = anP 2 ,
Preliminaries
By wallpaper group we mean a group G of
Euclidean isometries of the real plane whose
translational subgroup is a lattice generated by
two linearly independent translations. As is well
known, there are seventeen isomorphism classes
of such groups. Also well known is the fact that
if ρ is a rotation in one of these groups, then its
order is 2, 3, 4, or 6.
Unlike many discussions of patterns in the
plane which refer to subsets, called motifs, being
repeated without overlap [4, p. 204], we use analysis to develop the concept of pattern. For us, a
pattern is given by a wallpaper function, which is
a real- (or perhaps complex-) valued function on
the real plane that is invariant under the action
of one of the wallpaper groups [2]. In symbols, we
require that
f (gx) = f (x) for every x ∈ R2 , g ∈ G.
For any given group G, it is easy to construct such functions by superimposing plane
waves invariant with respect to the lattice of
translations in the group. My paper “Wallpaper
functions” [2] explains the construction and also
covers functions with color-reversing symmetries.
For wallpaper functions with 3-fold symmetry, an
alternate method is possible, one that readers familiar with group representations may recognize.
We present this method and try to generalize it
to produce functions with 5-fold symmetry. In the
generalization, we can see exactly why Figure 1
fails to enjoy honest translational symmetry.
Constructing Patterns with 3-fold Symmetry
To construct functions on the plane with 3-fold
symmetry, we start with an unlikely object: cyclic
permutation of three variables, considered as
a linear transformation of R3 . After all, this
permutation, which we will call P , does have
order 3. If f is any function of three variables,
then f (x, y, z) + f (z, x, y) + f (y, z, x) is a function
invariant under P .
Since we seek periodic functions, let us suppose
that the function f (x, y, z) is periodic with respect
to the integer lattice in R3 . Let us limit ourselves
to constructing continuously differentiable functions. Then we may assume that f is represented
as the sum of its Fourier series in three variables:
X
f (x, y, z) =
an e2π in·x .
n∈Z3
November 2012
For 3-fold symmetry, we may either take such
an f and average it over cyclic permutation of
variables or require that
since these coefficients will become equal after
averaging. (Here, we think of the matrix P as
acting on row vectors n, the adjoint action.)
It remains to find planar functions somewhere
in this 3-dimensional setup. The eigenspaces of
the linear transformation P are easily found to be
the line generated by [1, 1, 1] and a plane Π, with
basis V1 = [1, −1, 0] and V2 = [0, 1, −1], on which
P acts as rotation through an angle 2π /3.
If we restrict f to the plane Π, the resulting
function has translational invariance with respect
to the integer vectors V1 and V2 , as well as
rotational symmetry through an angle of 2π /3.
Of course, f is necessarily invariant with respect
to all compositions of these symmetries and so
is a wallpaper function with group at least as
large as the one generated by the rotation and one
translation. (Translation along V2 can be written by
conjugating the other translation by the rotation.)
We call this group p3, preferring the notation
of the International Union of Crystallography to
Conway’s orbifold notation.
We can easily construct wallpaper functions
with groups that contain p3 as a subgroup by
including special groupings of terms in the Fourier
expansion of the function of three variables before
we restrict. To explain this in some detail, we begin
with packets of waves invariant under 3-fold
rotation:
1
2
Wn (x) = (e2π in·x + e2π in·P x + e2π in·P x ),
3
where we divide by three simply so that Wn (0) = 1.
Notice that when restricted to the plane Π,
Wn+(j,j,j) = Wn for j ∈ N.
Setting n = (n1 , n2 , n3 ) and representing a typical point in Π as XV1 + Y V2 , we write these wave
packets in the form
Wn,m (X, Y ) =
1 2π i(nX+mY )
(e
+ e2π i(mX−(n+m)Y )
3
+ e2π i(−(n+m)X+nY )),
where
n = n1 − n2 and m = n2 − n3 .
If we wish to choose n and m first and find a
vector n that gives rise to those frequencies, one
choice is (n + m, m, 0).
This is why we claim that every (sufficiently
smooth) wallpaper function on the plane with 3fold rotational symmetry can be exhibited as the
restriction of a function of three variables that is
periodic with respect to the integer lattice and invariant under cyclic permutation of variables: The
Notices of the AMS
1387
functions Wn,m (X, Y ) form a basis for functions
with the desired symmetry in the plane, and every
Wn,m is the restriction of some Wn . We write
X
an,m Wn,m (X, Y )
(1)
f (X, Y ) =
n,m
for the typical wallpaper function—a superposition of wallpaper waves.
To continue the discussion of functions with
additional symmetries, we omit some messy details and present the reader with the fact that the
map
σc (X, Y ) = (Y , X)
is a mirror reflection in Π, so that any sum of the
form (1) where
an,m = am,n
will represent a function with mirror symmetry.
Every function of the form
X
f (X, Y ) =
an,m (Wn,m (X, Y ) + Wm,n (X, Y ))
n,m
is invariant under the group p31m (or 3*3 for
orbifold enthusiasts).
It is amusing to choose coefficients to create
pleasing functions. One example of a function
invariant under p31m is shown in Figure 2.
It is likewise amusing to work out recipes
to create functions invariant under the groups
p3m1, p6, and p6m, but that information appears
elsewhere [2], along with the details about various
ways to use color to depict a complex-valued
function in the plane, so we move on to 5-fold
symmetry.
Before we do, note this crucial fact about the
construction: We were able to find a basis for
Π consisting of vectors with integer coordinates.
This is the only reason we could claim that the
restricted function is periodic with respect to a
lattice within Π, given only that it is periodic with
respect to the integer lattice in R3 .
What Happens If We Use 5 Variables?
The same construction can be attempted in R5 . Let
us review why we do not expect to create functions
with 5-fold symmetry that are also periodic with
respect to a lattice. The reason to include this wellknown explanation is that it suggests something
to look for in the images.
The Crystallographic Restriction
We prove a special case of the crystallographic
restriction, showing that 5-fold rotations are never
present in wallpaper groups: Suppose a wallpaper
group G contains a translation T and a rotation R
through 2π /5 radians. In any wallpaper group, one
may always find a shortest translation, so suppose
further that T is a shortest translation in G. It is
easy to check that U := RT R −1 and V := R −1 T R
are translations along vectors at angles of ±2π /5
radians from the direction of T , as shown in
Figure 3. If T translates the center of rotation O
to the point A, then the composite translation UV
translates O to X, producing a shorter translation
and contradicting our assumptions. Therefore, no
wallpaper group contains a rotation of order 5.
-----→
- ------→
It is easy to compute
√ that the ratio OA/OX is
the golden ratio, (1 + 5)/2, which we denote φ.
The reader may enjoy looking back at Figure 1,
which seems to be invariant under a translation
T and a rotation R through 2π /5 radians. Where
is the shorter translation that our computations
have guaranteed?
A
T
X
U
R
V
U
O
Figure 2. A function with symmetry group p31m.
1388
Figure 3. A translation and a rotation through
72◦ create a shorter translation: The
composition of V := R −1 T R with U := RT R −1
produces a translation that is shorter than T by
a factor of 1/φ
1/φ.
Notices of the AMS
Volume 59, Number 10
Wallpaper with 5-fold Symmetry?
Let us imitate the procedure we used for 3-fold
symmetry. Again we call P the linear transformation defined by cyclic permutation of the variables:
P (x, y, z, u, v) = (v, x, y, z, u).
Likewise, we take f to be any function periodic
with respect to the integer lattice in R5 and
symmetric under cyclic permutation. An easy
first example, which we will carry through the
remainder of this section, is
f (x, y, z, u, v) = sin(2π x) + sin(2π y) + sin(2π z)
+ sin(2π u) + sin(2π v).
The eigenspaces of P are one line spanned
by [1, 1, 1, 1, 1] and two planes, with P acting
as rotation through 2π /5 in one plane and
4π /5 in the other. We select the first of these
planes to call Π. One basis for Π is
E1 = [1, cos(2π //5), cos(4π/5), cos(6π /5), cos(8π /5)],
E2 = [0, sin(2π /5), sin(4π /5), sin(6π /5), sin(8π /5)],
In our construction of functions with 3-fold
symmetry we easily found an eigenbasis with
integer entries. Some reorganization of E1 and E2
gives the basis
1
1
1 1
,
, − 2 , − , 0],
φ φ2
φ
φ
V1
=
[
V2
=
[0,
1
1
1 1
,
, − 2 , − ] = P V1 ,
φ φ2
φ
φ
where φ is the golden ratio.
We can see that there is no way to obtain integer
entries from these: If we clear a denominator of
φ2 , we create a factor of φ in another entry. In
fact, this plane is irrational, in the sense that it
contains no rational vectors at all.
This throws a fly into the ointment if we
are trying to construct wallpaper functions with
5-fold symmetry. However, it leads us to interesting pictures. Let us start with a function invariant
under P and defined by a Fourier series relative to
the integer lattice in R5 . Then let us restrict f to Π
by the equation
F(s, t) = f (sV1 + tV2 ).
When we do this with our example of the cyclic
sum of sines, we obtain the function pictured in
Figure 4. The origin of Π, near the lower left of the
picture, is, in fact, a center of 5-fold symmetry.
Toward the top there seems to be a translated
appearance of the same pattern, with what looks
like another 5-center. Another appears on a ray
through the origin 72◦ from the vertical. Divide
either of these apparent translation vectors by φ
and notice that there is, well, not translational
symmetry, but translational ballpark nearness of
the pattern. Divide again.
November 2012
Figure 4. A cyclic sum of sine waves, restricted
to the plane Π .
To move in the other direction, we recompute
the image for the translated function f ((s+89)V1 +
tV2 ). It looks exactly the same as Figure 4! We
know that this function cannot have translational
symmetry, but something interesting is going on.
Perhaps you have already guessed what, having
recognized the number 89. Applying the Binet
formula for Fn , the nth Fibonacci number,
φn − (−φ)−n
√
,
Fn =
5
we find that, although there are no integer vectors
in the plane Π, there are integer vectors arbitrarily
close:
Fn V1 = [Fn−1 , Fn−2 , −Fn−2 , −Fn−1 , 0]
+ (−φ)−n [−1, 1, −1, 1, 0],
and likewise for Fibonacci multiples of V2 . From
this, it is an easy estimate to find that
|F(s + Fn , t) − F(s, t)| < 5/φn .
Although this exemplary function f cannot
have any translational symmetries (we know this
from the crystallographic restriction), it does have
what we will call quasisymmetries. As with Bohr’s
almost-periodic functions, given any ǫ > 0, we can
find a translational distance T so that
|F(s + T , t) − F(s, t)| < ǫ.
Conclusion
The example is only one in a large space of
functions with the same symmetry, or rather
quasisymmetry. Just as in our construction of
functions with 3-fold symmetry, we can start
with any cyclically invariant waves in 5-space and
Notices of the AMS
1389
Superimposing only cosine waves does create
functions with symmetry about five mirror axes,
at least at one point. The opening example enjoyed
this extra symmetry.
Though we have not violated the crystallographic restriction, we have found an interesting
family of functions. They invite our eye to wander
and enjoy the near-repeats. As for the attractive
fraud that opened this paper: A bit of Photoshopping made it look rather more symmetrical
than it is. Figure 5 gives an unaltered view of the
quasisymmetry.
References
Figure 5. An unretouched depiction of our effort
to create 5 -fold symmetry.
restrict them to the plane Π. There are limited
possibilities for mirror symmetry, though we will
mention that the example that arose from the sine
function had a mirror color-reversing symmetry.
[1] Daniel Cleary, Once-ridiculed discovery redefined
the term crystal, Science 335 (Oct. 14, 2011), p. 165.
[2] Frank Farris, Wallpaper functions, Expositiones
Math. 20:3 (2002), 1–22.
, Vibrating wallpaper, Communications in
[3]
Visual Mathematics 1 (1998) (now vol. 0 of JOMA).
[4] Branko Grünbaum and G. C. Shephard, Tilings and
Patterns, W. H. Freeman and Co., New York, 1987.
[5] Marjorie Senechal, Quasicrystals and Geometry,
Cambridge University Press, Cambridge, UK, 1995.
[6]
, What is a quasicrystal, Notices Amer. Math.
Soc. 53:8 (2006), 886–887.
About the Cover
Irrational Symmetry
The cover image was produced by Frank Farris, and is a modification of Figure 4
in his article in this issue.
It records the values (through coloring and shading) of a fairly simple periodic
function (specified in that article), restricted to one of the eigenplanes of the cyclic
permutation in R5 . What’s important here, as he says in the article, is that although
crystalline 5-fold symmetry is impossible, it can be very closely approximated. The
basic idea behind what one sees is already apparent in two dimensions—the figure
below shows in a similar way the restriction of cos(x) cos(y) to an irrational line in
R2 .
But the details of what happens in Farris’ figure are nonetheless striking, perhaps even hypnotic, and what one sees is certainly more striking in two dimensions
than in one. Mathematically, the effect is due to lattice points in Z5 that are near
to the plane, and the extraordinary accuracy with which patterns are repeated is
presumably due to the accuracy with which the golden ratio is tightly approximated by rational numbers. And perhaps also something intrinsic to human visual
perception.
We thank Frank Farris for the time and effort spent on this.
—Bill Casselman
Graphics Editor
([email protected])
1390
Notices of the AMS
Volume 59, Number 10